Factorization presentations

TL;DR

This paper introduces factorization presentations to demonstrate that sheaves of coinvariants over moduli spaces of stable pointed curves are coherent under finiteness conditions on vertex operator algebra modules, removing the need for semisimplicity.

Contribution

It introduces the concept of factorization presentations and proves coherence of sheaves of coinvariants without assuming semisimplicity of the VOA modules.

Findings

Sheaves of coinvariants are coherent on moduli spaces under finiteness conditions.

Factorization presentations provide a new framework for analyzing these sheaves.

Coherence holds without the semisimplicity assumption.

Abstract

Modules over a vertex operator algebra V give rise to sheaves of coinvariants on moduli of stable pointed curves. If V satisfies finiteness and semi-simplicity conditions, these sheaves are vector bundles. This relies on factorization, an isomorphism of spaces of coinvariants at a nodal curve with a finite sum of analogous spaces on the normalization of the curve. Here we introduce the notion of a factorization presentation, and using this, we show that finiteness conditions on V imply the sheaves of coinvariants are coherent on moduli spaces of pointed stable curves without any assumption of semisimplicity.